3.536 \(\int (e x)^{5/2} (a+b x^3)^{5/2} (A+B x^3) \, dx\)

Optimal. Leaf size=404 \[ -\frac{27\ 3^{3/4} a^{11/3} e^2 \sqrt{e x} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt{\frac{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\sqrt [3]{a}+\left (1+\sqrt{3}\right ) \sqrt [3]{b} x\right )^2}} (4 A b-a B) \text{EllipticF}\left (\cos ^{-1}\left (\frac{\sqrt [3]{a}+\left (1-\sqrt{3}\right ) \sqrt [3]{b} x}{\sqrt [3]{a}+\left (1+\sqrt{3}\right ) \sqrt [3]{b} x}\right ),\frac{1}{4} \left (2+\sqrt{3}\right )\right )}{11264 b^2 \sqrt{\frac{\sqrt [3]{b} x \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\sqrt [3]{a}+\left (1+\sqrt{3}\right ) \sqrt [3]{b} x\right )^2}} \sqrt{a+b x^3}}+\frac{81 a^3 e^2 \sqrt{e x} \sqrt{a+b x^3} (4 A b-a B)}{5632 b^2}+\frac{27 a^2 (e x)^{7/2} \sqrt{a+b x^3} (4 A b-a B)}{1408 b e}+\frac{15 a (e x)^{7/2} \left (a+b x^3\right )^{3/2} (4 A b-a B)}{704 b e}+\frac{(e x)^{7/2} \left (a+b x^3\right )^{5/2} (4 A b-a B)}{44 b e}+\frac{B (e x)^{7/2} \left (a+b x^3\right )^{7/2}}{14 b e} \]

[Out]

(81*a^3*(4*A*b - a*B)*e^2*Sqrt[e*x]*Sqrt[a + b*x^3])/(5632*b^2) + (27*a^2*(4*A*b - a*B)*(e*x)^(7/2)*Sqrt[a + b
*x^3])/(1408*b*e) + (15*a*(4*A*b - a*B)*(e*x)^(7/2)*(a + b*x^3)^(3/2))/(704*b*e) + ((4*A*b - a*B)*(e*x)^(7/2)*
(a + b*x^3)^(5/2))/(44*b*e) + (B*(e*x)^(7/2)*(a + b*x^3)^(7/2))/(14*b*e) - (27*3^(3/4)*a^(11/3)*(4*A*b - a*B)*
e^2*Sqrt[e*x]*(a^(1/3) + b^(1/3)*x)*Sqrt[(a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2)/(a^(1/3) + (1 + Sqrt[3])*
b^(1/3)*x)^2]*EllipticF[ArcCos[(a^(1/3) + (1 - Sqrt[3])*b^(1/3)*x)/(a^(1/3) + (1 + Sqrt[3])*b^(1/3)*x)], (2 +
Sqrt[3])/4])/(11264*b^2*Sqrt[(b^(1/3)*x*(a^(1/3) + b^(1/3)*x))/(a^(1/3) + (1 + Sqrt[3])*b^(1/3)*x)^2]*Sqrt[a +
b*x^3])

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Rubi [A]  time = 0.34375, antiderivative size = 404, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.192, Rules used = {459, 279, 321, 329, 225} \[ \frac{81 a^3 e^2 \sqrt{e x} \sqrt{a+b x^3} (4 A b-a B)}{5632 b^2}-\frac{27\ 3^{3/4} a^{11/3} e^2 \sqrt{e x} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt{\frac{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\sqrt [3]{a}+\left (1+\sqrt{3}\right ) \sqrt [3]{b} x\right )^2}} (4 A b-a B) F\left (\cos ^{-1}\left (\frac{\left (1-\sqrt{3}\right ) \sqrt [3]{b} x+\sqrt [3]{a}}{\left (1+\sqrt{3}\right ) \sqrt [3]{b} x+\sqrt [3]{a}}\right )|\frac{1}{4} \left (2+\sqrt{3}\right )\right )}{11264 b^2 \sqrt{\frac{\sqrt [3]{b} x \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\sqrt [3]{a}+\left (1+\sqrt{3}\right ) \sqrt [3]{b} x\right )^2}} \sqrt{a+b x^3}}+\frac{27 a^2 (e x)^{7/2} \sqrt{a+b x^3} (4 A b-a B)}{1408 b e}+\frac{15 a (e x)^{7/2} \left (a+b x^3\right )^{3/2} (4 A b-a B)}{704 b e}+\frac{(e x)^{7/2} \left (a+b x^3\right )^{5/2} (4 A b-a B)}{44 b e}+\frac{B (e x)^{7/2} \left (a+b x^3\right )^{7/2}}{14 b e} \]

Antiderivative was successfully verified.

[In]

Int[(e*x)^(5/2)*(a + b*x^3)^(5/2)*(A + B*x^3),x]

[Out]

(81*a^3*(4*A*b - a*B)*e^2*Sqrt[e*x]*Sqrt[a + b*x^3])/(5632*b^2) + (27*a^2*(4*A*b - a*B)*(e*x)^(7/2)*Sqrt[a + b
*x^3])/(1408*b*e) + (15*a*(4*A*b - a*B)*(e*x)^(7/2)*(a + b*x^3)^(3/2))/(704*b*e) + ((4*A*b - a*B)*(e*x)^(7/2)*
(a + b*x^3)^(5/2))/(44*b*e) + (B*(e*x)^(7/2)*(a + b*x^3)^(7/2))/(14*b*e) - (27*3^(3/4)*a^(11/3)*(4*A*b - a*B)*
e^2*Sqrt[e*x]*(a^(1/3) + b^(1/3)*x)*Sqrt[(a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2)/(a^(1/3) + (1 + Sqrt[3])*
b^(1/3)*x)^2]*EllipticF[ArcCos[(a^(1/3) + (1 - Sqrt[3])*b^(1/3)*x)/(a^(1/3) + (1 + Sqrt[3])*b^(1/3)*x)], (2 +
Sqrt[3])/4])/(11264*b^2*Sqrt[(b^(1/3)*x*(a^(1/3) + b^(1/3)*x))/(a^(1/3) + (1 + Sqrt[3])*b^(1/3)*x)^2]*Sqrt[a +
b*x^3])

Rule 459

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_)), x_Symbol] :> Simp[(d*(e*x)^(m
+ 1)*(a + b*x^n)^(p + 1))/(b*e*(m + n*(p + 1) + 1)), x] - Dist[(a*d*(m + 1) - b*c*(m + n*(p + 1) + 1))/(b*(m +
n*(p + 1) + 1)), Int[(e*x)^m*(a + b*x^n)^p, x], x] /; FreeQ[{a, b, c, d, e, m, n, p}, x] && NeQ[b*c - a*d, 0]
&& NeQ[m + n*(p + 1) + 1, 0]

Rule 279

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[((c*x)^(m + 1)*(a + b*x^n)^p)/(c*(m +
n*p + 1)), x] + Dist[(a*n*p)/(m + n*p + 1), Int[(c*x)^m*(a + b*x^n)^(p - 1), x], x] /; FreeQ[{a, b, c, m}, x]
&& IGtQ[n, 0] && GtQ[p, 0] && NeQ[m + n*p + 1, 0] && IntBinomialQ[a, b, c, n, m, p, x]

Rule 321

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(c^(n - 1)*(c*x)^(m - n + 1)*(a + b*x^n
)^(p + 1))/(b*(m + n*p + 1)), x] - Dist[(a*c^n*(m - n + 1))/(b*(m + n*p + 1)), Int[(c*x)^(m - n)*(a + b*x^n)^p
, x], x] /; FreeQ[{a, b, c, p}, x] && IGtQ[n, 0] && GtQ[m, n - 1] && NeQ[m + n*p + 1, 0] && IntBinomialQ[a, b,
c, n, m, p, x]

Rule 329

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> With[{k = Denominator[m]}, Dist[k/c, Subst[I
nt[x^(k*(m + 1) - 1)*(a + (b*x^(k*n))/c^n)^p, x], x, (c*x)^(1/k)], x]] /; FreeQ[{a, b, c, p}, x] && IGtQ[n, 0]
&& FractionQ[m] && IntBinomialQ[a, b, c, n, m, p, x]

Rule 225

Int[1/Sqrt[(a_) + (b_.)*(x_)^6], x_Symbol] :> With[{r = Numer[Rt[b/a, 3]], s = Denom[Rt[b/a, 3]]}, Simp[(x*(s
+ r*x^2)*Sqrt[(s^2 - r*s*x^2 + r^2*x^4)/(s + (1 + Sqrt[3])*r*x^2)^2]*EllipticF[ArcCos[(s + (1 - Sqrt[3])*r*x^2
)/(s + (1 + Sqrt[3])*r*x^2)], (2 + Sqrt[3])/4])/(2*3^(1/4)*s*Sqrt[a + b*x^6]*Sqrt[(r*x^2*(s + r*x^2))/(s + (1
+ Sqrt[3])*r*x^2)^2]), x]] /; FreeQ[{a, b}, x]

Rubi steps

\begin{align*} \int (e x)^{5/2} \left (a+b x^3\right )^{5/2} \left (A+B x^3\right ) \, dx &=\frac{B (e x)^{7/2} \left (a+b x^3\right )^{7/2}}{14 b e}-\frac{\left (-14 A b+\frac{7 a B}{2}\right ) \int (e x)^{5/2} \left (a+b x^3\right )^{5/2} \, dx}{14 b}\\ &=\frac{(4 A b-a B) (e x)^{7/2} \left (a+b x^3\right )^{5/2}}{44 b e}+\frac{B (e x)^{7/2} \left (a+b x^3\right )^{7/2}}{14 b e}+\frac{(15 a (4 A b-a B)) \int (e x)^{5/2} \left (a+b x^3\right )^{3/2} \, dx}{88 b}\\ &=\frac{15 a (4 A b-a B) (e x)^{7/2} \left (a+b x^3\right )^{3/2}}{704 b e}+\frac{(4 A b-a B) (e x)^{7/2} \left (a+b x^3\right )^{5/2}}{44 b e}+\frac{B (e x)^{7/2} \left (a+b x^3\right )^{7/2}}{14 b e}+\frac{\left (135 a^2 (4 A b-a B)\right ) \int (e x)^{5/2} \sqrt{a+b x^3} \, dx}{1408 b}\\ &=\frac{27 a^2 (4 A b-a B) (e x)^{7/2} \sqrt{a+b x^3}}{1408 b e}+\frac{15 a (4 A b-a B) (e x)^{7/2} \left (a+b x^3\right )^{3/2}}{704 b e}+\frac{(4 A b-a B) (e x)^{7/2} \left (a+b x^3\right )^{5/2}}{44 b e}+\frac{B (e x)^{7/2} \left (a+b x^3\right )^{7/2}}{14 b e}+\frac{\left (81 a^3 (4 A b-a B)\right ) \int \frac{(e x)^{5/2}}{\sqrt{a+b x^3}} \, dx}{2816 b}\\ &=\frac{81 a^3 (4 A b-a B) e^2 \sqrt{e x} \sqrt{a+b x^3}}{5632 b^2}+\frac{27 a^2 (4 A b-a B) (e x)^{7/2} \sqrt{a+b x^3}}{1408 b e}+\frac{15 a (4 A b-a B) (e x)^{7/2} \left (a+b x^3\right )^{3/2}}{704 b e}+\frac{(4 A b-a B) (e x)^{7/2} \left (a+b x^3\right )^{5/2}}{44 b e}+\frac{B (e x)^{7/2} \left (a+b x^3\right )^{7/2}}{14 b e}-\frac{\left (81 a^4 (4 A b-a B) e^3\right ) \int \frac{1}{\sqrt{e x} \sqrt{a+b x^3}} \, dx}{11264 b^2}\\ &=\frac{81 a^3 (4 A b-a B) e^2 \sqrt{e x} \sqrt{a+b x^3}}{5632 b^2}+\frac{27 a^2 (4 A b-a B) (e x)^{7/2} \sqrt{a+b x^3}}{1408 b e}+\frac{15 a (4 A b-a B) (e x)^{7/2} \left (a+b x^3\right )^{3/2}}{704 b e}+\frac{(4 A b-a B) (e x)^{7/2} \left (a+b x^3\right )^{5/2}}{44 b e}+\frac{B (e x)^{7/2} \left (a+b x^3\right )^{7/2}}{14 b e}-\frac{\left (81 a^4 (4 A b-a B) e^2\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{a+\frac{b x^6}{e^3}}} \, dx,x,\sqrt{e x}\right )}{5632 b^2}\\ &=\frac{81 a^3 (4 A b-a B) e^2 \sqrt{e x} \sqrt{a+b x^3}}{5632 b^2}+\frac{27 a^2 (4 A b-a B) (e x)^{7/2} \sqrt{a+b x^3}}{1408 b e}+\frac{15 a (4 A b-a B) (e x)^{7/2} \left (a+b x^3\right )^{3/2}}{704 b e}+\frac{(4 A b-a B) (e x)^{7/2} \left (a+b x^3\right )^{5/2}}{44 b e}+\frac{B (e x)^{7/2} \left (a+b x^3\right )^{7/2}}{14 b e}-\frac{27\ 3^{3/4} a^{11/3} (4 A b-a B) e^2 \sqrt{e x} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt{\frac{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\sqrt [3]{a}+\left (1+\sqrt{3}\right ) \sqrt [3]{b} x\right )^2}} F\left (\cos ^{-1}\left (\frac{\sqrt [3]{a}+\left (1-\sqrt{3}\right ) \sqrt [3]{b} x}{\sqrt [3]{a}+\left (1+\sqrt{3}\right ) \sqrt [3]{b} x}\right )|\frac{1}{4} \left (2+\sqrt{3}\right )\right )}{11264 b^2 \sqrt{\frac{\sqrt [3]{b} x \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\sqrt [3]{a}+\left (1+\sqrt{3}\right ) \sqrt [3]{b} x\right )^2}} \sqrt{a+b x^3}}\\ \end{align*}

Mathematica [C]  time = 0.178099, size = 116, normalized size = 0.29 \[ \frac{e^2 \sqrt{e x} \sqrt{a+b x^3} \left (7 a^3 (a B-4 A b) \, _2F_1\left (-\frac{5}{2},\frac{1}{6};\frac{7}{6};-\frac{b x^3}{a}\right )-\left (a+b x^3\right )^3 \sqrt{\frac{b x^3}{a}+1} \left (7 a B-28 A b-22 b B x^3\right )\right )}{308 b^2 \sqrt{\frac{b x^3}{a}+1}} \]

Antiderivative was successfully verified.

[In]

Integrate[(e*x)^(5/2)*(a + b*x^3)^(5/2)*(A + B*x^3),x]

[Out]

(e^2*Sqrt[e*x]*Sqrt[a + b*x^3]*(-((a + b*x^3)^3*Sqrt[1 + (b*x^3)/a]*(-28*A*b + 7*a*B - 22*b*B*x^3)) + 7*a^3*(-
4*A*b + a*B)*Hypergeometric2F1[-5/2, 1/6, 7/6, -((b*x^3)/a)]))/(308*b^2*Sqrt[1 + (b*x^3)/a])

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Maple [C]  time = 0.059, size = 5063, normalized size = 12.5 \begin{align*} \text{output too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((e*x)^(5/2)*(b*x^3+a)^(5/2)*(B*x^3+A),x)

[Out]

result too large to display

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (B x^{3} + A\right )}{\left (b x^{3} + a\right )}^{\frac{5}{2}} \left (e x\right )^{\frac{5}{2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x)^(5/2)*(b*x^3+a)^(5/2)*(B*x^3+A),x, algorithm="maxima")

[Out]

integrate((B*x^3 + A)*(b*x^3 + a)^(5/2)*(e*x)^(5/2), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left ({\left (B b^{2} e^{2} x^{11} +{\left (2 \, B a b + A b^{2}\right )} e^{2} x^{8} +{\left (B a^{2} + 2 \, A a b\right )} e^{2} x^{5} + A a^{2} e^{2} x^{2}\right )} \sqrt{b x^{3} + a} \sqrt{e x}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x)^(5/2)*(b*x^3+a)^(5/2)*(B*x^3+A),x, algorithm="fricas")

[Out]

integral((B*b^2*e^2*x^11 + (2*B*a*b + A*b^2)*e^2*x^8 + (B*a^2 + 2*A*a*b)*e^2*x^5 + A*a^2*e^2*x^2)*sqrt(b*x^3 +
a)*sqrt(e*x), x)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x)**(5/2)*(b*x**3+a)**(5/2)*(B*x**3+A),x)

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (B x^{3} + A\right )}{\left (b x^{3} + a\right )}^{\frac{5}{2}} \left (e x\right )^{\frac{5}{2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x)^(5/2)*(b*x^3+a)^(5/2)*(B*x^3+A),x, algorithm="giac")

[Out]

integrate((B*x^3 + A)*(b*x^3 + a)^(5/2)*(e*x)^(5/2), x)